1. Field of the Invention
The present invention relates to adaptive equalization methods and apparatus. More specifically, the present invention relates to methods and apparatus for equalizing a dispersive channel.
2. Description of the Related Art
Filtering is a common and powerful function that finds use in a variety of applications. One application is in communication systems in which information is sent from one place to another. When an applied filter is used to compensate for the effects of the channel across which the information is sent, the filter is typically referred to as an equalizer.
A major source of error in information transmission is intersymbol interference (ISI) that arises when a signal is sent across a dispersive channel. Dispersive channels tend to spread the energy of a transmitted signal out over time, which means both past and future symbols can interfere with the current symbol.
To further illustrate this point, consider a transmitted signal x[k] which is sent across a dispersive channel with an impulse response h[k]. The received signal, y[k], is given by:
                                                                        y                ⁡                                  [                  k                  ]                                            =                                                ∑                  n                                ⁢                                                                  ⁢                                  h                  ⁡                                      [                    n                    ]                                                  ⁢                                  x                  ⁡                                      [                                          k                      -                      n                                        ]                                                                                                                          =                                                                    h                    ⁡                                          [                      0                      ]                                                        ⁢                                      x                    ⁡                                          [                      k                      ]                                                                      +                                                      ∑                                          n                      <                      0                                                        ⁢                                                                          ⁢                                                            h                      ⁡                                              [                        n                        ]                                                              ⁢                                          x                      ⁡                                              [                                                  k                          -                          n                                                ]                                                                                            +                                                      ∑                                          n                      >                      0                                                        ⁢                                                                          ⁢                                                            h                      ⁡                                              [                        n                        ]                                                              ⁢                                                                  x                        ⁡                                                  [                                                      k                            -                            n                                                    ]                                                                    .                                                                                                                              (        1        )            The second term in equation (1) arises from the precursor component of the channel impulse response and allows future symbols to interfere with the current symbol. The third term in equation (1) arises from the post-cursor component of the channel impulse response and allows previous symbols to interfere with the current symbol. Fortunately, equalization can be used to reduce or remove these components.
Adaptive Transversal Filters
Oftentimes, one has little or no prior knowledge of the channel characteristics, making it difficult to define an appropriate filter. To overcome this problem, filters are often made adaptive, allowing them to “learn” the channel characteristics. One type of filter used in adaptive equalization applications is the adaptive transversal filter. Adaptive transversal filters are well understood, non-recursive structures that operate in the discrete time-domain and have a finite impulse response (FIR).
FIG. 1 is a block diagram of a conventional adaptive transversal filter 100 including a tapped delay line 102 coupled to N+1 number of multipliers 103. The tapped delay line 102 comprises N delay blocks 104 each configured to delay an input signal x[k] by one sample delay time Z−1. The input signal x[k] and the sample-time delayed signals x[k−1], x[k−2], . . . , x[k−N] are multiplied by respective weights or filter coefficients W0, W1, W2, . . . , WN and summed to produce a filter output signal y[k]. For convenience, the input signal x[k] and the sample time delayed signals x[k−1], x[k−2], . . . x[k−N] as well as the filter coefficients W0, W1, W2, . . . , WN are expressed respectively as vectors:Xk=[x[k] x[k−1] . . . x[k−N]]T  (2),andWk=[W0[k] W1[k] . . . WN[k]]T  (3),where T denotes vector transpose.
The filter output signal y[k] is provided to an adaptation engine 106 configured to automatically adjust the filter coefficients W0, W1, W2, . . . , WN based on a desired response d[k] as compared to the filter output signal y[k]. Typically, the desired response d[k] is stored in a receiver and includes a copy of a known sequence transmitted to the receiver during a training mode.
The adaptation engine 106 includes an adaptation algorithm configured to update the filter coefficients W0, W1, W2, . . . , WN with time. Commonly used adaptation algorithms attempt to reduce the mean square error E[εk2], where the error signal εk is given by:εk=d[k]−y[k]=d[k]−WkTXk  (4).Expanding the square of the error signal gives:
                                                                        ɛ                k                2                            =                                                (                                                            d                      ⁡                                              [                        k                        ]                                                              -                                                                  W                        k                        T                                            ⁢                                              X                        k                                                                              )                                2                                                                                        =                                                                    d                    ⁡                                          [                      k                      ]                                                        2                                +                                                      W                    k                    T                                    ⁢                                      X                    k                                    ⁢                                      X                    k                    T                                    ⁢                                      W                    k                                                  -                                  2                  ⁢                                      d                    ⁡                                          [                      k                      ]                                                        ⁢                                      X                    k                    T                                    ⁢                                                            W                      k                                        .                                                                                                          (        5        )            
To produce a reasonably simplified expression for the mean-square error, several assumptions may be made:                Wk is fixed; and        Xk, d[k], and εk are statistically wide-sense stationary.With these assumptions, the mean-square error reduces to:        
                              E          ⁡                      [                          ɛ              k              2                        ]                          =                              E            ⁡                          [                                                d                  ⁡                                      [                    k                    ]                                                  2                            ]                                +                                    W              T                        ⁢                          E              ⁡                              [                                                      X                    k                                    ⁢                                      X                    k                    T                                                  ]                                      ⁢            W                    -                      2            ⁢                          E              ⁡                              [                                                      d                    ⁡                                          [                      k                      ]                                                        ⁢                                      X                    k                    T                                                  ]                                      ⁢                          W              .                                                          (        6        )            
From equation (6), it is clear that the mean-square error E[εk2] is a quadrative function of the coefficient vector W. This quadratic function is referred to as the “error surface” and contains a global minimum at an optimal coefficient vector W. The task of the adaptation engine 106 is to walk the filter coefficients W0, W1, W2, . . . , WN down the error surface to a point close to the optimal solution.
There are a variety of basic algorithms available to converge the coefficient vector Wk towards the optimal solution, including Newton's method, the steepest descent method, least-mean square (LMS) method, and recursive least squares (RLS) method. LMS is a commonly used algorithm due to its ease of computation. LMS achieves its simplicity by approximating the mean-square error E[εk2] with εk2, leading to the following coefficient update equation:Wk+1=Wk+μεkXk  (7),where μ is a step-size scalar that can be used to control convergence rate and steady-state accuracy.
Local Minima on the Error Surface
Under certain circumstances, local minima can also exist on the error surface. Adaptation engine that become trapped on a local minimum provide a non-optimal coefficient vector W, which reduces the effectiveness of the transversal filter. Local minima are typically caused by non-linear effects in the signal path, certain channel characteristics, or combinations thereof.
Blind Equalization
When the desired response d[k] is unknown, adaptation is typically done in blind mode. There are many algorithms capable of blindly converging an adaptive filter. Typically, algorithms suitable for blind equalization use higher-order statistics of the filter's input. Example algorithms include Sato's algorithm and the Constant Modulus Algorithm (CMA).
Decision Feedback Equalizers
The Decision Feedback Equalizer (DFE) is an alternative to the feedforward transversal filter. Adaptive DFEs typically use adaptive transversal filters, such as the adaptive transversal filter 100 shown in FIG. 1, in a feedback role. DFEs can also use adaptive transversal filters in a feedforward role. In a typical symbol-rate DFE, precursor and post-cursor components spaced at integer multiples of the symbol period TS are corrected for. For example, a DFE with N feedback taps can correct for post-cursor components that occur at spacings of TS, 2TS, . . . , NTS from the current symbol. DFEs can be implemented in analog or digital form. Digital implementations typically use analog-to-digital conversion of the filter's input signal.
FIG. 2 is a block diagram of a conventional adaptive DFE 200 having a feedforward section 202, a feedback section 204 and a decision device 206. The feedforward section 202 includes a first transversal filter 208 and a first adaptation engine 210 configured to receive a feedforward training signal dff[k]. The role of the feedforward section 202 is to reduce the precursor component of the ISI. The feedback section 204 includes a second transversal filter 212 and a second adaptation engine 214 configured to receive a feedback training signal dfb[k]. The role of the feedback section 204 is to reduce the post-cursor component of the ISI.
FIG. 3 is a block diagram of a conventional decision-directed DFE 300 including a feedforward transversal filter 302, a decision device 304, a feedback transversal filter 306 and an adaptation engine 308. The decision-directed DFE 300 operates in a “decision-directed” mode in which the Output signal 310 of the decision device 304 is used to create the desired response provided to the adaptation engine 308. Specifically, an error signal εk comprising the difference between the input 314 of the decision device 304 and the Output signal 310 is provided to the adaptation engine 308 as the desired response. Thus, the decision-directed DFE 300 does not require a training signal to converge the adaptation engine 308. However, convergence is more difficult.
The decision-directed DFE 300 shown in FIG. 3 uses a common error signal εk and adaptation engine 308 to adapt both the feedforward transversal filter 302 and feedback transversal filter 306. The generation of the error signal εk can be challenging and uses additional conditioning circuitry 314 to process the soft decisions (i.e., the input 314 of the decision device 304) before subtracting them from the hard decisions (i.e., the Output signal 310). The conditioning circuit 314 can include, for example, sample-and-hold circuitry, automatic gain control circuitry, combinations of the foregoing, or the like. The conditioning circuitry 314 accounts for the delay through the decision device 304 and also prevents the hard decisions from swamping the small signal level of the soft decisions.
Fractionally Spaced Equalizers
Fractionally Spaced Equalizers (FSEs) are transversal equalizers that include taps spaced at some fraction of the symbol period TS. FSEs are used, for example, as a linear equalizer or the feedforward portion of a DFE. A typical choice for tap spacing is TS/2, which allows correction of both the in-phase components and the quadrature components in the channel impulse response.
For an ideal, jitter-free sampling clock, equalization of anything but the ideal in-phase samples provides no improvement in performance. However, when a realistic, jittered clock is considered, the true sampling instant slides around the ideal point. Thus, FSEs that provide equalization across the symbol period provide improved performance to symbol-rate equalizers.